Quarks at the modular $S_4$ cusp

We analyse the possibility of describing quark masses, mixing and CP violation in $S_4^{\mathrm{\prime }}$ modular flavour models without flavons. We focus on the case where the closeness of the modulus to the point of residual ${\mathbb{Z}}_3^{ST}$ symmetry (the cusp) plays a role in generating quark mass hierarchies and discuss the role modular form normalisations play in such constructions. We find that fitting quark data requires explicit CP breaking, unless a second modulus is introduced.